Ingeometry, theunit hyperbola is the set of points (x,y) in theCartesian plane that satisfy theimplicit equation${\displaystyle x^2-y^2=1.}$ In the study ofindefinite orthogonal groups, the unit hyperbola forms the basis for analternative radial length

${\displaystyle r=\sqrt{x^2-y^2}.}$

Whereas theunit circle surrounds its center, the unit hyperbola requires theconjugate hyperbola${\displaystyle y^2-x^2=1}$ to complement it in the plane. This pair ofhyperbolas share theasymptotesy =x andy = −x. When the conjugate of the unit hyperbola is in use, the alternative radial length is${\displaystyle r=\sqrt{y^2-x^2}.}$

The unit hyperbola is a special case of therectangular hyperbola, with a particularorientation,location, andscale. As such, itseccentricity equals${\displaystyle \sqrt{2}.}$^[1]

The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction ofspacetime as apseudo-Euclidean space. There the asymptotes of the unit hyperbola form alight cone. Further, the attention to areas ofhyperbolic sectors byGregoire de Saint-Vincent led to thelogarithm function and the modern parametrization of the hyperbola by sector areas. When the notions of conjugate hyperbolas and hyperbolic angles are understood, then the classicalcomplex numbers, which are built around the unit circle, can be replaced with numbers built around the unit hyperbola.

Generally asymptotic lines to a curve are said to converge toward the curve. Inalgebraic geometry and the theory ofalgebraic curves there is a different approach to asymptotes. The curve is first interpreted in theprojective plane usinghomogeneous coordinates. Then the asymptotes are lines that are tangent to the projective curve at apoint at infinity, thus circumventing any need for a distance concept and convergence. In a common framework (x, y, z) are homogeneous coordinates with theline at infinity determined by the equationz = 0. For instance, C. G. Gibson wrote:^[2]

For the standard rectangular hyperbola${\displaystyle f=x^2-y^2-1}$  in${\displaystyle {\mathbb{R}}^2}$ , the corresponding projective curve is${\displaystyle F=x^2-y^2-z^2,}$  which meetsz = 0 at the pointsP = (1 : 1 : 0) andQ = (1 : −1 : 0). BothP andQ aresimple onF, with tangentsx +y = 0,x −y = 0; thus we recover the familiar 'asymptotes' of elementary geometry.

TheMinkowski diagram is drawn in a spacetime plane where the spatial aspect has been restricted to a single dimension. The units of distance and time on such a plane are

• units of 30 centimetres length andnanoseconds, or
• astronomical units and intervals of 8 minutes and 20 seconds, or
• light years andyears.

Each of these scales of coordinates results inphoton connections of events along diagonal lines ofslope plus or minus one.Five elements constitute the diagramHermann Minkowski used to describe the relativity transformations: the unit hyperbola, itsconjugate hyperbola, the axes of the hyperbola, a diameter of the unit hyperbola, and theconjugate diameter.The plane with the axes refers to a restingframe of reference. The diameter of the unit hyperbola represents a frame of reference in motion withrapiditya where tanha =y/x and (x,y) is the endpoint of the diameter on the unit hyperbola. The conjugate diameter represents thespatial hyperplane of simultaneity corresponding to rapiditya.In this context the unit hyperbola is acalibration hyperbola^[3][4]Commonly in relativity study the hyperbola with vertical axis is taken as primary:

The arrow of time goes from the bottom to top of the figure — a convention adopted byRichard Feynman in his famous diagrams. Space is represented by planes perpendicular to the time axis. The here and now is a singularity in the middle.^[5]

The vertical time axis convention stems from Minkowski in 1908, andis also illustrated on page 48 of Eddington'sThe Nature of the Physical World (1928).

A direct way to parameterizing the unit hyperbola starts with the hyperbolaxy = 1 parameterized with theexponential function:${\displaystyle (e^t,e^{-t}).}$ 

This hyperbola is transformed into the unit hyperbola by alinear mapping having the matrix 

${\displaystyle (e^t,e^{-t})A=(\frac{e^t+e^{-t}}{2},\frac{e^t-e^{-t}}{2})=(\cosh t,\sinh t).}$ 

This parametert is thehyperbolic angle, which is theargument of thehyperbolic functions.

One finds an early expression of the parametrized unit hyperbola inElements of Dynamic (1878) byW. K. Clifford. He describes quasi-harmonic motion in a hyperbola as follows:

The motion${\displaystyle \rho =\alpha \cosh (nt+ϵ)+\beta \sinh (nt+ϵ)}$  has some curious analogies to elliptic harmonic motion. ... The acceleration${\displaystyle \ddot{\rho }=n^2\rho ;}$  thus it is always proportional to the distance from the centre, as in elliptic harmonic motion, but directedaway from the centre.^[6]

As a particularconic, the hyperbola can be parametrized by the process of addition of points on a conic. The following description was given by Russian analysts:

Fix a pointE on the conic. Consider the points at which the straight line drawn throughE parallel toAB intersects the conic a second time to be thesum of the points A and B.

For the hyperbola${\displaystyle x^2-y^2=1}$  with the fixed pointE = (1,0) the sum of the points${\displaystyle (x_1,y_1)}$  and${\displaystyle (x_2,y_2)}$  is the point${\displaystyle (x_1{x}_2+y_1{y}_2,y_1{x}_2+y_2{x}_1)}$  under the parametrization${\displaystyle x=\cosh t}$  and${\displaystyle y=\sinh t}$  this addition corresponds to the addition of the parametert.^[7]

Whereas the unit circle is associated withcomplex numbers, the unit hyperbola is key to thesplit-complex number plane consisting ofz =x +yj, wherej² = +1.Thenjz = y + xj, so the action ofj on the plane is to swap the coordinates. In particular, this action swaps the unit hyperbola with its conjugate and swaps pairs ofconjugate diameters of the hyperbolas.

In terms of thehyperbolic angle parametera, the unit hyperbola consists of points

${\displaystyle \pm (\cosh a+j\sinh a)}$ , wherej = (0,1).

The right branch of the unit hyperbola corresponds to the positive coefficient. In fact, this branch is the image of theexponential map acting on thej-axis. Thus this branch is the curve${\displaystyle f(a)=\exp (aj).}$  Theslope of the curve ata is given by thederivative

${\displaystyle f^{\mathrm{\prime }}(a)=\sinh a+j\cosh a=jf(a).}$  For anya,${\displaystyle f^{\mathrm{\prime }}(a}$ ) ishyperbolic-orthogonal to${\displaystyle f(a)}$ . This relation is analogous to the perpendicularity of exp(a i) and i exp(a i) when i² = − 1.

Since${\displaystyle \exp (aj)\exp (bj)=\exp ((a+b)j)}$ , the branch is agroup under multiplication.

Unlike thecircle group, this unit hyperbola group isnotcompact.Similar to the ordinary complex plane, a point not on the diagonals has apolar decomposition using the parametrization of the unit hyperbola and the alternative radial length.

• F. Reese Harvey (1990)Spinors and calibrations, Figure 4.33, page 70,Academic Press,ISBN 0-12-329650-1 .